Farb–Wolfson–Wood motivic stabilization conjecture

Let $\mathcal U$ be a geometrically connected and smooth ungraded variety. For $\in\mathbb Z_{\geq 0}^{r}$, let $\mathcal Z_{,n}$ be the locus of ordered $r$-tuples of effective zero-cycles of multidegree

whose greatest common divisor is $n$-powerfree, and let $\mathcal X_{}$ denote the corresponding product of symmetric products. Let $\widehat{\mathcal M}_{\mathcal X}$ be the indicated localization of the completed Grothendieck ring, and let $Z_{\mathcal U}$ be the motivic \zeta function. **Farb–Wolfson–Wood motivic stabilization conjecture.** As the entries of

tend to infinity at any rates,

$$\frac{[\mathcal Z_{,n}]}{[\mathcal X_{}]}$$

converges in $\widehat{\mathcal M}_{\mathcal X}$ to

$$Z_{\mathcal U}(\mathbb L^{-rn\dim\mathcal U})^{-1}.$$

This is a graded generalization of motivic stabilization for configuration spaces and relates the asymptotic classes of tuples of cycles with bounded common divisor to the motivic zeta function.